Definition (homotopy image)

A morphism f:c→df : c \to d in CC is called a homotopy monomorphism if the universal morphism Id×Id:c→c×dhcId \times Id : c \to c \times^h_d c into its homotopy pullback along itself is an isomorphism in the homotopy category.

The homotopy image of ff is a factorization of ff into a cofibration c→f(c)c \to f(c) followed by a homotopy monomorphism f(c)→df(c) \to d

such that for any other such factorization c→e→dc \to e \to d there exists a unique morphism f(c)→ef(c) \to e in the homotopy category making the obvious triangles commute.